Möbius transformation copies the annulus $\{z:r<|z|<1\}$ to the domain between $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$
Please help me to find what is $r$.
2026-04-06 11:21:55.1775474515
Möbius transformation question
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Hint: The inverse of this transform is also Möbius and might for simplicity map $1\mapsto 1$, $-1\mapsto-1$, $\frac12\mapsto r$, $0\mapsto -r$. Can you find $r$ in this special case? Can you justify the simplification?